Some new applications of the pointwise relations for the relative rearrangement

Abstract

Let $V=V(\Omega)$ be a set satisfying the Poincaré-Sobolev pointwise inequalities for the relative rearrangement and $\rho$ an arbitrary norm on the set of measurable functions on the interval $\Omega_*=(0,measure(\Omega))$. Then using the associate norm of $\rho$, we give some sufficient conditions to ensure that a function $u$ of $V$ has to be bounded or integrable in an Orlicz space. We show similar results for a solution of quasilinear equation under some conditions between the data and $\rho$. We then give an unified approach for many kinds of estimates. Using a main relation involving some pointwise relations for the relative rearrangement, we prove a regularity result for the time derivative of a parabolic system well posed in a Grand Sobolev space.